Warrington presents an interesting paper wherein middle-school aged students from all backgrounds find solutions to given fractions without the aid of any 'tricks' or techniques. And they do so with accuracy and without a lack of speed. Indeed, Warrington's class demonstrates the benefits of non-algorithmic learning in that "responses were forthcoming", and "[children could take] a straight computation problem and assign meaning to it by creating a word problem." Fractions seem to make more sense this way, and they have meaning. I also think that by invoking continual classroom discussion like Warrington has done in her classroom that the children will not be so timid to ask questions in the future and are more willing to other people ideas, other points of view for the same problem thus expounding upon their own knowledge and making it stronger.
In contrast however, students without any set of rules they can refer to result in (more often than not) nonefficient work and prone to occasional failure. Warrington admits that it's a lengthy process when she explains that for the understanding of a single problem (4 2/5 divided by 1/3) "went on for some time, and for many it carried over into recess." Also, without a way to check one's answer like you'd be able to with an algorithm or two, answers are less consistent. The obvious example is that given the same problem stated above, the children in Warrington's classroom all got the answer wrong except for one. So although the children were undoubtedly smart, they didn't have a solid way of doing fractions, only ways they came up with themselves.
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I wasn't quite able to follow your summary. However I do agree with continual classroom discussion because it does in deed help students get out of their "shell" and helps them to be able to be more open. this helps the teacher better understnad where they are at in their learning.
ReplyDeleteI liked your response very much. I thought you gave a great summary. I question the very last sentence because it was placed in the disadvantage paragraph. Is coming up with their own ways by themselves a disadvantage? And maybe a solid way of doing them isn't great because different things work for everyone. I think if it's what they came up with, it must work for them and make sense to them, which I think is a good thing. Great job!
ReplyDeleteI agree with the advantages.
ReplyDeleteTo me, the two paragraphs didn't seem to relate to each other.
Students, with or without, set algorithms will make mistakes. In fact, I think it's a lot more likely for a student using a meaningless algorithm to make a mistake, than someone who can reason through the problem. I have helped a lot of people who get confused when plugging things into the equation or have no idea why it would be different in a different context. I think this is where the error is more likely to occur. When it does occur under these circumstances, how is the student supposed to check their work, because it makes no sense to them. The only thing they can do is go through the computations again. Whereas, a student who reasons can guesstimate a reasonable answer, and they also will have alternate ways to check their work.
I really liked the advantage you stated about being able to ask questions without reservation. I know that I was not afraid to do this growing up, and I learned so much more from it. Now though, I am a tutor for some younger kids, and the one thing they really struggle with is asking when they don't understand. If they were able to do this, they would be able to understand concepts so much more quickly.
ReplyDeleteOne thing that I think a little different on is the fact that a disadvantage is the time. Something I have learned is the more I think about something, the more I come to understand it. Yes, having time limits is sometimes helpful, but I loved nothing more than feeling accomplished after I knew that I learned something so solid, that I could reproduce it at any given time.